Grothendieck, the great and strange mathematician, complained about the nature of mathematics:
"I think there's an inherent flaw in the mathematical way of thinking and I have the impression that it's also reflected in this Manichaean vision of human nature. On the one hand, there's the good, and on the other, the bad, and in the best case scenario, we see both living side by side."
Grothendieck, disappointed, has left mathematics and went into
seclude in Pyrenees. I do not understand it. Personally I do not see
anything wrong with mathematics, and with yes-no Aristotelian logic:
There is good and there is evil and there is the specific situation that determines which is which. What's wrong with this?
From The Guardian article about Alexander Grothedieck of Aug 31, 2024: "He was in mystic delirium’: was this hermit mathematician a forgotten genius whose ideas could transform AI – or a lonely madman?"
"I open a first page at random. The writing is spidery; there are
occasional multicoloured topological diagrams, namechecks of past
thinkers, often physicists – Maxwell, Planck, Einstein – and recurrent
references to Satan and “this cursed world”
whose ideas could transform AI
So I stay with the devil of the algebra and the angel of geometry, and with their happy marriage - geometric algebra. Which geometric algebra? This is a good question. Certainly time is not the same as space. Einstein and Minkowski merged space with time, but their reasoning was highly questionable. Maxwell equations were taken as a starting point. 10-parameters Poincare group came out rather easily as an invariance group, and it was set on a pedestal. It was given the absolute power over all other possible interactions, for no good reason. Then it was discovered that Maxwell equations are invariant under the 15-parameters conformal group, containing the Poincare group as a subgroup leaving the "conformal infinity" (Dupin cyclide) as the unmoved "absolute", but it didn't help us with understanding the nature of time. Is time "real" or "imaginary"? Or both? There is a lot that needs to be done in physics before jumping ahead into the future of fancy mathematics with no "yes-no" dichotomy. Some physicists promote the view that the universe is a quantum computer - like it would explain anything. It does not explain anything at all, it just sweeps problems under the carpet, since we do not understand what quantum theory is about. Quantum theory does not explain consciousness. We need to understand what "consciousness", "information", and "measurements" are first. Then, perhaps, we will be able to get some idea about quantum theory, the theory of "measurements".
Let's go back to algebra, geometric algebra. The mystery of spin is
hidden there. We do not need to mix space and time from the start (see Peter Woit blog "The Mystery of Spin"). If
time comes out of space and spin - that would be a nice surprise.
Geometric algebra A = Cl(V) of space happens to be
isomorphic to complex quaternions that can be faithfully represented by
2×2 complex matrices Mat(2,C). Unexpected meeting of
geometry with the formalism of quantum mechanics. Coincidence? Or there
is some deep meaning hidden in this fact? Can we read/decode the
message?
Hamilton invented quaternions when trying to make space into an
algebra. He was forced to add the fourth dimension. What is the meaning
of this fourth dimension? Is it just pure mathematics that has nothing
to do with reality? Certainly the fourth dimension of Hamilton is not
the "real time" of special relativity theory. So what is it? For a while
we have a toy that we are playing with, discovering new
functionalities. I talked to Laura about all these mysteries, and she
told me that as a little girl she owned a doll that could even pee. Of
course you had first to fill the doll with water. What can our Cl(V)
produce if properly filled with proper stuff? Shall we get spinors?
Feed 1.
First we have fed A with an orthogonal projector p = (1+n)/2, p = p* = p2. It produced the right eigensubspace
{u: up=u} = Ap,
which is a non-trivial left ideal of A. It carried an irreducible representation of A, where "spinors" live. We also got its orthogonal complement A(1-p). Spinors can reside also there.
Feed 2.
Then we have chosen a different approach, through the Gelfand-Neumark-Segal (GNS) construction. This time we fed A with a "state" - a positive linear functional f on A. It produced a left ideal If:
If = {u: f(u*u)=0}
which we didn't like (as it consists of norm zero vectors), so we have taken the quotient Hilbert space Hf = A/If, build as the set of equivalence classes [a], and we got a representation ρf of A on Hf with cyclic vector Ωf = [1] such that
f(a) = (Ωf,ρf(a)Ωf).
If f is a pure state, then the representation ρf is irreducible (we did not show it yet). So elements of Hf are also candidates for spinors. The relation to Feed 1 is still waiting for an explanation.
Feed 3.
Here again we start with a state f, but this time we add some salt - we use the fact that our A is not only a *-algebra, but also a Hilbert space on its own. This allows us to represent f through an algebra element, denoted by the same letter, so that
f(a) = <f*, a>. (1)
Taking the positive square root f½ of f we get
f(a) = <f½, L(a)f½ >,
where L is the left regular representation of A on A.
Connecting Feed 2 to Feed 3.
We can now happily use Proposition 4.5.3 from the GNS Promenade
to connect the two constructions. For this to take place, Feed 3 should
result in a cyclic representation. As it stands, the representation is
not necessarily cyclic in A. Let us understand this
problem better. Let us look at Eq. (1), but from the right to the left.
If f, on the right, is a positive element of the algebra A, then (1) defines a positive functional. What are the simplest possible positive elements of A? The answer is: they are orthogonal projections p=p*=p2. So, what happens if we take f = p = (1+n)/2? Since p = p* = p2, it follows that f½ = p. In that case L(A)f½ = Ap is our left ideal of Feed 1. It is certainly not the whole A - for this choice of f. Thus, in this case
f½ is not cyclic in A. But it is cyclic in the left ideal Af½ ! Define H'f = L(A)f½,
Ω'f = f½, ρ'f = L restricted to H'f. Then, according to Proposition 4.5.3, there exists an isometry U from Hf to H'f, that maps Ωf into
f½, and U intertwines the two representations:
Uρf(a) = L(a)U on Hf.
The
GNS construction (Feed 2) is not really needed in our case. The point
is that GNS construction is quite general, it works with any C*-algebra.
Here we have a particular case, when our algebra is already a Hilbert
space (perhaps too big), so we can apply Feed 3.
We have already made a connection to Feed 1 by taking f = p, but this needs to be made clear. We will do it in the next post.
As for spinors - they are lurking, always nearby, but still not as main characters.
P.S. 04-02-25 From my correspondence: "... Allais' effect only worked when his pendulum was triggered by a man. Allais' effect was officially recognized by the French academy of sciences."
Interesting. Quoting from Wikipedia
https://en.wikipedia.org/wiki/Allais_effect
Maurice Allais states that the eclipse effect is related to a gravitational anomaly that is inexplicable in the framework of the currently admitted theory of gravitation, without giving any explanation of his own.[30] Allais's explanation for another anomaly (the lunisolar periodicity in variations of the azimuth of a pendulum) is that space evinces certain anisotropic characteristics, which he ascribes to motion through an aether which is partially entrained by planetary bodies.
His hypothesis leads to a speed of light dependent on the moving direction with respect to a terrestrial observer, since the Earth moves within the aether but the rotation of the Moon induces a "wind" of about 8 km/s. Thus Allais rejects Einstein's interpretation of the Michelson–Morley experiment and the subsequent verification experiments of Dayton Miller.[31][32]
